Rodgers is said to have held talks with the Bhoys this week, having left Anfield in October.
Rodgers, who was sacked by the Reds in October, was reported to have held talks with the Bhoys on Tuesday, having emerged as the club's preferred choice to replace the departing Ronny Deila.
Former Cardiff City boss Malky Mackay is also said to have held talks, while David Moyes and Roy Keane remain among the bookies' favourites.
The Scottish champions have since issued a statement confirming they have spoken to a number of candidates, but deny they have a preferred candidate for the position.
However, Liverpool will be hoping it is Rodgers who gets the gig, as it would save them £10 million, according to the Record.
The Scottish tabloid claims the Northern Irishman stands to net £5 million a year over the next two years as part of a severance package agreed upon his departure.
But it is said that payments will cease as soon as Rodgers signs a contract elsewhere, meaning Celtic may have to dig deep to lure the 43-year-old out of unemployment.